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Additive reduced basis preconditioners for large-scale parametrized PDEs

Xinnan Fan, Qixiao Hu, Shiquan Zhang

arXiv:2609.11097v1math.NAcs.NA

TL;DR

Direct reduced-basis solves can be limited when small spaces do not approximate demanding PDE solutions adequately. The paper adds correction terms to reduced-basis preconditioners and dynamically constructs their reduced spaces for FGMRES. Across the tested PDE problems, the authors report robust convergence and online speed-ups, with practical applicability to repeated solves.

  • Problem

    Direct reduced-basis solves can require larger spaces and more expensive offline construction when solution manifolds are difficult to approximate to strict tolerances.

    Source: §1 · p. 2

  • Method

    ARB preconditioners add correction terms to the reduced-basis approximation and use iteration-dependent POD spaces within FGMRES for full-order solves.

    Source: §1 · p. 2, §3.3 · p. 8

  • Results

    Across convection-diffusion, anisotropic vortex, Stokes, and Helmholtz tests, ARB preconditioners show robust convergence under parameter variations and significant online speed-ups.

    Source: §5 · p. 21

  • Takeaways & Limitations

    Low break-even points in the experiments support the practical applicability of ARB preconditioners to multi-query problems requiring repeated high-fidelity evaluations.

    Source: §5 · p. 21

    The Type I preconditioner may be singular for some choices of α, so its admissibility is checked numerically.

    Source: §3.1 · p. 7, §3.2 · p. 7

Abstract

from arXiv · show

We introduce a class of additive reduced basis preconditioners designed to accelerate the iterative solution of large-scale linear systems arising from discretized parametrized PDEs. The main idea is to regularize the inherently singular reduced-order approximation by adding simple correction terms: either a scaled identity correction or a projected correction based on a basic preconditioner. This yields nonsingular preconditioners under explicit and easily checked conditions. The construction and application of the preconditioners are integrated into an FGMRES framework through an offline strategy that dynamically builds the reduced-basis component by proper orthogonal decomposition at each FGMRES step. We establish sufficient conditions for the nonsingularity of the preconditioners and derive error bounds for the preconditioned Richardson iteration. Numerical results for convection-diffusion, anisotropic vortex, Stokes, and Helmholtz problems are provided to verify the efficiency and convergence of the proposed ARB preconditioners. The method consistently converges in a few iterations and substantially reduces online solve time, supporting its efficiency for multi-query engineering scenarios.

1. Introduction.

Reduced-order models can ease repeated PDE solves, but direct reduced solves may require large spaces and expensive offline construction when solutions are difficult to approximate. The paper proposes additive reduced-basis preconditioners for full-order iterative solves, combining reduced-space information with correction terms.

1. Introduction.

  • Slow Kolmogorov-width decay in transport-dominated and wave problems can make small static reduced spaces miss strict tolerances, while larger spaces weaken the savings through greater offline cost.Local nonlinear reduced bases may mitigate this limitation, but achieving the required accuracy can still require a much larger reduced space.Source: §1 · p. 2
  • Projection-based reduced-basis preconditioning injects snapshot information into Krylov solvers while retaining the full-order residual as the convergence criterion.The introduction situates this approach alongside Krylov deflation, POD-augmented iterations, and coarse-space preconditioning.Source: §1 · p. 2
  • ARB preconditioners add simple correction terms to a singular reduced-basis approximation, with explicit nonsingularity conditions and inexpensive application formulas.They are combined with FGMRES to solve the full-order model, so final accuracy is that of the high-fidelity discretization rather than the reduced-basis approximation alone.Source: §1 · p. 2
  • The paper presents experiments on convection-diffusion, anisotropic vortex, Stokes, and Helmholtz problems, comparing ARB methods with ILU and AMG.The paper is organized to cover RB and GMRES preliminaries, ARB construction and dynamic generation, numerical experiments, and practical implications.Source: §1 · p. 2

2. Preliminaries.

The preliminaries formulate the discretized parametrized PDE and review reduced-basis and GMRES tools. They explain how POD builds a reduced space from snapshots and motivate using that space as a preconditioning component to accelerate iterative solves toward the desired accuracy.

2. Preliminaries.

  • The discretized parametrized problem is a linear system with parameter-dependent stiffness matrix A_h(µ), assumed nonsingular on the working space.The reduced-basis method approximates its solution manifold in a subspace of dimension N much smaller than the full dimension N_h.Source: §2 · p. 3, §2.1 · p. 3

2.1. The RB method.

  • POD forms a reduced basis from high-fidelity training snapshots using leading left singular vectors, with the dimension selected by the decay of singular values and a relative-energy tolerance.The Galerkin projection yields a reduced system; when parameter dependence is affine, offline-online decomposition enables rapid online assembly and solution.Source: §2.1 · pp. 3–4

2.2. The GMRES method.

  • GMRES minimizes the residual over an affine Krylov subspace, using Arnoldi to reduce the minimization to a low-dimensional least-squares problem.As iterations grow, storage and orthogonalization costs increase; restarting bounds the Krylov-subspace dimension.Source: §2.2 · p. 4
  • GMRES may converge slowly without a problem-dependent preconditioner, while an RB method alone can incur high offline costs when high accuracy is needed.The paper therefore uses RB as a GMRES preconditioning component to reduce offline burden while accelerating the iterative scheme.Source: §2.2 · pp. 4–5, §3 · p. 5

3. ARB preconditioners.

The paper constructs additive reduced-basis preconditioners to regularize a singular reduced-order approximation, then builds iteration-dependent bases for use with FGMRES. It establishes nonsingularity conditions and analyzes convergence with Richardson iteration.

3.1. Construction of ARB preconditioners.

  • ARB regularizes the singular RB inverse approximation with additive terms: αI (Type I) or a projected basic-preconditioner correction (Types II/III).The correction is intended to ensure nonsingularity while the reduced-basis component supplies the approximation.Source: §3.1 · pp. 5–6
  • Theorem 3.4 states that when A11 is nonsingular and T is definite (positive or negative), Type II and Type III preconditioners are nonsingular for any α ≠ 0; Type I is not guaranteed to be well defined.Source: §3.2 · p. 7, §3.1 · p. 7

3.3. Dynamic construction of ARB preconditioners.

  • Because solution snapshots do not ensure a uniformly accurate operator approximation, the method builds iteration-dependent reduced bases from correction snapshots using POD.The construction adds a simple correction term to ensure nonsingularity and is presented as easier to understand and implement than the related MSRB construction.Source: §3.3 · p. 8

3.4. ARB preconditioners for Richardson method.

  • Richardson analysis relates prescribed projection or error tolerances to convergence; greedy construction can enforce its hypotheses, while POD controls aggregate snapshot projection error.The paper also states that a corresponding residual result holds.Source: §3.4 · pp. 9–10

3.5. ARB preconditioners for FGMRES.

  • FGMRES accommodates the iteration-dependent preconditioner, while its offline construction generates reduced spaces from recursively obtained snapshots after an initial large-scale solve.Training problems can be solved in parallel, and the final constructed basis is reused if FGMRES takes more iterations than the number of generated spaces.Source: §3.5 · pp. 10–11

4. Numerical experiments.

The numerical experiments evaluate dynamically constructed ARB-FGMRES preconditioners across four PDE test cases, comparing them with ILU and AMG over 100 random online solves and testing robustness on indefinite Helmholtz systems. The Helmholtz experiment also reports about 113 solves for k ∈ (5, 10].

4. Numerical experiments.

  • Unless stated otherwise, offline training parameters were sampled uniformly over prescribed domains; each reported online statistic used 100 randomly generated parameters distinct from offline samples.In the convection-dominated, SUPG-stabilized range µ ∈ [10−5, 2.25 × 10−3], training used L=6 dynamic reduced spaces from ns=50 logarithmically sampled parameters, and online tests used 100 independently and log-uniformly sampled parameters.Source: §4 · p. 12, §4.1 · p. 14

4.1. Case 1: The convection-diffusion equation.

  • For µ ∈[0.1, 1], ARB mean speed-ups were 26.1×–28.1× over ILU(2) and 3.1×–3.3× over AMG.In this range, ARB methods took about 3.5–3.6 iterations and 0.16–0.17 seconds per solve, compared with AMG’s 5.05 iterations and 0.525 seconds.Source: §4.1 · pp. 13–14, Table 1 · p. 13, Table 2 · p. 14
  • For µ ∈[10−5, 2.25 × 10−3], ARB mean speed-ups were 1.0×–1.1× over ILU(2) and 6.8×–7.7× over AMG.ARB required about 4.0 iterations and 0.217–0.245 seconds per solve, comparable to ILU(2) and substantially faster than AMG.Source: Table 1 · p. 13, Table 2 · p. 14, §4.1 · p. 14

4.2. Case 2: The anisotropic vortex problem.

  • For the anisotropic vortex problem, ARB methods converged in about 4 iterations and took 0.627–0.644 seconds online, with mean speed-ups of 1.6×–1.7× over ILU(3) and 1.6× over AMG.All minimum speed-ups over both baselines exceeded 1×.Source: Table 3 · p. 16, Table 4 · p. 16, §4.2 · pp. 16–17

4.3. Case 3: The Stokes equation.

  • Stokes ARB methods averaged 3.15, 3.27, and 2.21 iterations for Types I–III, with mean speed-ups of 165.8×–177.2× over ILU(2) and 26.8×–28.6× over AMG.The AMG baseline used a structure-preserving field split; direct application to the full saddle-point matrix diverged in these tests.Source: Table 5 · p. 18, Table 6 · p. 18, §4.3 · p. 18

4.4. Case 4: The Helmholtz equation.

  • For k ∈[1, 5], mean speed-ups were 17.9×–18.5× over ILU(3) and 2.9×–3.0× over AMG; for k ∈(5, 10], they were 46.9×–49.1× and 5.9×–6.1×, respectively.All three ARB preconditioners converged for every tested sample, with average iteration counts between 3.44 and 3.85 across the two ranges.Source: Table 8 · p. 20, §4.4 · p. 20, Table 7 · p. 20
  • For k ∈ (5, 10], the Helmholtz experiment reports about 113 solves.The supplied passage also contains an incomplete fragment citing [1, 5], so no comparison can be established from it.Source: §4.4 · p. 21
  • The proposed ARB preconditioners are dynamically constructed during the offline stage and applied within FGMRES.Source: §5 · p. 21
  • The analysis establishes nonsingularity under mild conditions and derives error bounds for preconditioned Richardson iteration.Source: §5 · p. 21

5. Conclusions.

Across four PDE problems, numerical experiments show robust ARB convergence and significant online speed-ups relative to standard preconditioners. Low break-even points support practical use in multi-query settings.

5. Conclusions.

  • Experiments on convection-diffusion, anisotropic vortex, Stokes, and Helmholtz problems demonstrate ARB effectiveness relative to ILU and AMG.Source: §5 · p. 21
  • ARB preconditioners show robust convergence across parameter variations and achieve significant online speed-ups.Source: §5 · p. 21
  • Low break-even points highlight ARB preconditioners’ practical applicability when high-fidelity models require repeated evaluations.Source: §5 · p. 21
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